In cancerpolicy/bcimodel: Model the impact of early detection and treatment interventions for breast cancer

knitr::opts_chunk$set(echo = TRUE)

Colombia, taking out endocrine only

propERpos <- 0.70
early5yr <- 0.87
adv5yr <- 0.55
propERTam <- 0.58

# Early stage
f = function(x, p=propERpos, t=propERTam, h=0.7, So=early5yr) { p*(1-t)*x + p*t*x^h + (1-p)*x - So}
str(Sb.early <- uniroot(f, lower=0.5, upper=early5yr, tol = 0.0001))

# Advanced stage 
f = function(x, p=propERpos, t=propERTam, h=0.7, So=adv5yr) { p*(1-t)*x + p*t*x^h + (1-p)*x - So}
str(Sb.adv <- uniroot(f, lower=0.2, upper=adv5yr, tol = 0.0001))

Colombia, taking out endocrine/chemo/combined

Treatments from the US in 2000 (see Annals paper):

Note: I could have detailed Endocrine, Chemo and Both for the ER-, but the table focuses on treatments that have an impact, grouped by the hazard ratio. So for ER-, chemo and both have the same HR, the HR for just chemo, since Endocrine has no effect in ER-.

ER Status | Stage | Treatment | Percent ---- | ------ | ----------- | ------ Positive | Advanced | Endocrine | 10 | | | Chemo | 38 | | | Both | 59 | | Early | Endocrine | 58 | | | Chemo | 8 | | | Both | 10 Negative | Advanced | Chemo OR both | 85+10 = 95 | | Early | Chemo OR both | 50+5 = 55

Description | Notation | Value --------------------------------- | -------- | --------------------- Proportion of all cases that are ER positive | $p_{i=ER+}$ | 0.70 Proportion of all cases that are ER negative | $p_{i=ER-} = 1-p_{j=ER+}$ | 0.30 Within stage and ER group i, proportion treated with treatment j | $t_{ij}$ | See table below Hazard ratio for treatment j | $h_{j}$ | See table below Observed 5-year survival, all ER | $S_{o}$ | 0.55 (advanced) or 0.87 (early) Baseline 5-year survival, all ER | $S_{b}$ | To be estimated for each stage

Using the assumption $S_{b}$ is the same for both ER+ and ER-, observed survival is sum of survivals under each treatment, weighted by the proportion receiving each treatment. For treatment $j$, survival is $(S_{b})^{h_{i}}$

$$S_{o} = \sum_{i}^{}p_{i}\sum_{j}^{} (t_{ij}S_{b})^{h_{j}}$$ Rearranging terms gives a fractional polynomial for which $S_{b}$ is the root:

$$0 = \sum_{i}^{}p_{i}\sum_{j}^{} (t_{ij}S_{b})^{h_{j}} - S_{o}$$

Stage | ER Status $i$ | Treatment $j$ | Hazard ratio $h_{j}$ | Proportion $t_{ij}$ ---- | ------ | ----------- | ------ | -------- Advanced | Positive | Endocrine | 0.7 | 10 | | | Chemo | 0.775 | 38 | | | Both | 0.5425 | 50 | | | None | 1 | 2 | | Negative | Endocrine | 1 | 3 | | | Chemo | 0.775 | 85 | | | Both | 0.775 | 10 | | | None | 1 | 2 Early | Positive | Endocrine | 0.7 | 58 | | | Chemo | 0.775 | 8 | | | Both | 0.5425 | 10 | | | None | 1 | 24 | | Negative | Endocrine | 1 | 20 | | | Chemo | 0.775 | 50 | | | Both | 0.775 | 5 | | | None | 1 | 25

For example, for the advanced stage in Colombia:

$$S_{o} = \sum_{i}^{}p_{i}\sum_{j}^{} (t_{ij}S_{b})^{h_{j}}$$ $$0.55 = 0.70(0.10{S_{b}}^{0.7} + 0.38{S_{b}}^{0.775} + 0.50{S_{b}}^{0.5425} + 0.02{S_{b}}^{1}) + 0.30(0.03{S_{b}}^{1} + 0.85{S_{b}}^{0.775} + 0.10{S_{b}}^{0.775} + 0.02{S_{b}}^{1}) $$ For the ER negative, no treatment and endocrine have the same hazard ratios, as do chemo and both. Those terms can thus be combined:

$$0.55 = 0.70(0.10{S_{b}}^{0.7} + 0.38{S_{b}}^{0.775} + 0.50{S_{b}}^{0.5425} + 0.02{S_{b}}^{1}) + 0.30(0.05{S_{b}}^{1} + 0.95{S_{b}}^{0.775}) $$

# Treated survivals
early5yr <- 0.87
adv5yr <- 0.55

# ER positivity
propERpos <- 0.70

# Treatment proportions: e=endocrine, c=chemo, b=both, pos=ER+, ''=ER-

# Hazard ratios: e=endocrine, c=chemo, b=both

# For chemo, the terms with the same hazards are combined, i.e. chemo and both are tc and none and endocrine are t

# Early stage
f = function(x, p=propERpos, tepos=0.58, tcpos=0.08, tbpos=0.10, tc=0.55, he=0.7, hc=0.775, So=early5yr) { 
  p*(1-tepos-tcpos-tbpos)*x + p*tepos*x^he + p*tcpos*x^hc + p*tbpos*x^(he*hc) + 
  (1-p)*(1-tc)*x  + (1-p)*tc*x^hc - So
  }
str(Sb.early <- uniroot(f, lower=0.5, upper=early5yr, tol = 0.0001))

# Advanced stage 
f = function(x, p=propERpos, tepos=0.10, tcpos=0.38, tbpos=0.59, tc=0.95, he=0.7, hc=0.775, So=adv5yr) { 
  p*(1-tepos-tcpos-tbpos)*x + p*tepos*x^he + p*tcpos*x^hc + p*tbpos*x^(he*hc) + 
  (1-p)*(1-tc)*x  + (1-p)*tc*x^hc - So
  }
str(Sb.adv <- uniroot(f, lower=0.2, upper=adv5yr, tol = 0.0001))

cancerpolicy/bcimodel documentation built on June 30, 2019, 12:39 a.m.